In addition, the following perturbed Rüssmann conditions hold. They are obtained by a perturbative argument from the equilibrium transversality conditions in Lemma  similarly to [1]} : for \(q_{0}\) , \(C_{0}\)  and \(\rho _{0}\)  as in Lemma , there exist \(\varepsilon _{0}>0\)  small enough such that for any \(\varepsilon \in [0,\varepsilon _{0}]\)  we have
